Ultrafinitism is pretty whacky since it requires tossing out the entire axiomatisation of mathematics. Ultrafinitism is inconsistent with axiomatisations even of fragments of Peano arithmetic (e.g. where you weaken the axiom scheme of induction to be restricted to a subset of formulae). You're left with something very cumbersome indeed. One thing the article fails to point out is that Gödel incompleteness still applies to such very weak theories. (It is actually misleading when talking about trying to prove the consistency of ZF - this too is not possible within ZF, provided ZF is consistent, due to incompleteness.)

There's a big difference between ultrafinitism and finitism - the latter allows a mathematical theory that allows you to talk indirectly about infinity but not directly. Peano arithmetic allows you to prove, for example, that, "for every prime number there is a larger one" - or as we might state it succinctly, "there are infinitely many prime numbers." But it doesn't allow you to talk about "the set of prime numbers" in the same direct way as you can talk about a single number or finite collection of numbers, and there is no such succinct way to say "there are infinitely many prime numbers" in the language of PA; you need the circumlocution.

These kinds of circumlocution are par for the course when dealing with weaker theories. But I think there is a huge practical issue with ultrafinitism which is exemplified thus: how do you prove a simple theorem in an ultrafinitist world, like for example, "the sum of any two even numbers is even" when it may be the case that the sum of two large even numbers is actually undefined? You have to write caveats all over every proof and I think for anything non-trivial it would swiftly become unmanageable.

On the philosophical side, I just don't think there is ever going to be a large number of people who think that "x+1 > x" is a statement that ought to be viewed with any level of suspicion. Mathematics is about creating abstractions from our real-world experience; numbers are such an abstraction. Real world objects are always finite, but that doesn't mean the abstraction has to be - to actually capture our intuition about how real world objects work, there can't be a limit.

[–] StillPaisleyCat@startrek.website 5 points 6 days ago (2 children)

At a certain point, I realized that from another perspective, the big divide seems to be between those who see continuous distributions as just an abstraction of a world that is inherently finite vs those who see finite steps as the approximation of an inherently continuous and infinitely divisible reality.

Since I’m someone who sees math as a way to tell internally-consistent stories that may or may not represent reality, I tend to have a certain exasperation with what seems to be the need of most engineers to anchor everything in Euclidean topography.

But it’s my spouse who had to help our kids with high school math. A parent who thinks non Euclidean geometry is fun is not helpful at that point.

[–] technocrit@lemmy.dbzer0.com 2 points 6 days ago* (last edited 6 days ago)

those who see continuous distributions as just an abstraction of a world that is inherently finite vs those who see finite steps as the approximation of an inherently continuous and infinitely divisible reality.

How about neither? Math is a formal system (like a game). It has no inherent relationship to "reality" or physics. There are only a few small areas of math that have been convincingly used in physical models, while the vast majority of mathematics is completely unrelated and even counter to physical assumptions (eg tarski paradox). Questions about the finiteness or divisibility of "reality" are scientific, not mathematical. Etc.

[–] FishFace@lemmy.world 2 points 6 days ago (1 children)

Yeah there is an important difference there. I think though that it's not clear whether the world is fundamentally discrete or continuous. As far as I know there is no evidence either way on this (though I remember reading that space and time must have the same discreteness/continuousness).

[–] technocrit@lemmy.dbzer0.com 1 points 6 days ago (1 children)

I think though that it’s not clear whether the world is fundamentally discrete or continuous.

Or both, neither, something else, etc.

[–] FishFace@lemmy.world 1 points 6 days ago

Not sure what that would even mean.

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